Question

Prove that $(cos x+cos y)^2+(sin x – sin y)^2 = 4 cos^2\,\frac{x+y}{2}$

Answer 1

L.H.S. = (cos x+ cos y)2+(sin x-sin y)2

=cos2x+cos2y+2cosx cosy+sin2 x+sin2 y-2 sinx sin y

=(cos2x+sin2x)+(cos2y+sin2y)+2(cosx cosy-sinx siny)

=1+1+2 cos(x+y) [cos(A+B)=(cos A cosB-sin A sin B)]

=2+2 cos(x+y)

=2[1+cos(x+y)]

$=2[1+2 cos^2(\frac{x+y}{2}-1]\,[cos2A=2cos^2 A-1]$

$=4cos^2(\frac{x+y}{2})=R.H.S.$